Choose the integer 2. Can its powers generate every nonzero residue modulo infinitely many primes? Artin’s primitive root conjecture predicts yes. OpenAI’s October 4, 2026 manuscript Primitive roots for every admissible integer base claims an unconditional proof, covering every integer except −1 and perfect squares. Its quantitative guarantee is a lower bound of order x/(logx)2x/(\log … Read more
A square of quantum spins can contain a vast number of possible states. The striking question is how much of that complexity a ground state actually uses to connect the square with its surroundings. OpenAI’s September 24, 2026 manuscript presents a proof of the 2D area law: under fixed locality bounds, a unique ground state … Read more
For 52 years, the Subset sum problem had a stubborn exponential benchmark: split the inputs in half and search at the 2n/22^{n/2} scale. OpenAI’s October 4, 2026 preprint presents a randomized classical algorithm running in O(20.49n)O(2^{0.49n}) for polynomial-bit inputs in a specified word-RAM model. It promises at least 2/32/3 correctness probability for every fixed input, … Read more
Ten pieces should have been enough. OpenAI’s September 23, 2026 preprint gives a compact set in nine-dimensional Euclidean space that refuses to fit that rule: every cover by ten subsets leaves at least one subset with the original diameter. The Borsuk conjecture counterexample lowers the previously reported counterexample dimension from 63 to nine. The original … Read more
Prime factors can be counted, but their parity has resisted a deceptively simple prediction for decades. The two-point Chowla conjecture asks whether the even-or-odd factor counts of two integers a fixed distance apart become uncorrelated. OpenAI’s September 24, 2026 manuscript claims a proof for ordinary averages at every cutoff, with a power-of-logarithm saving. It also … Read more
The Fast Fourier Transform has spent decades turning quadratic calculations into manageable work. OpenAI now presents a proof that discrete Fourier transform complexity can fall below the familiar O(nlogn)O(n\log n) scale, at every input length. The improvement is asymptotic, deterministic, and exact, but it belongs to a computational model with unrestricted complex coefficients and supplied … Read more
Comparing two strings sounds like a small job. Then the strings become long, insertions shift everything sideways, and a simple comparison becomes a search through millions of possible alignments. OpenAI’s September 24, 2026 manuscript tackles that cost with an edit distance approximation that combines almost-linear expected runtime with accuracy arbitrarily close to exact, provided the … Read more